1. Field of the Invention
The present invention relates to a method for vibration analysis according to the preamble of claim 1.
2. Background of the Invention
Modal identification is the process of estimating modal parameters from vibration measurements obtained from different locations of a structure. The modal parameters of a structure include the mode shapes, natural (or resonance) frequencies and the damping properties of each mode that influence the response of the structure in a frequency range of interest.
Modal parameters are important because they describe the inherent dynamic properties of the structure. Since these dynamics properties are directly related to the mass and the stiffness, experimentally obtained modal parameters provide information about these two physical properties of a structure. The modal parameters constitute a unique information that can be used for model validation, model updating, quality control and health monitoring.
In traditional modal analysis the modal parameters are found by fitting a model to the Frequency Response Function relating excitation forces and vibration response. In output-only modal analysis, the modal identification is performed based on the vibration responses only and a different identification strategy has to be used.
Output-only modal testing and analysis is used for civil engineering structures and large mechanical structures or structures m operation that are not easy to excite artificially.
Large civil engineering stares are not easily excited and they are often loaded by natural (ambient) loads that cannot easily be controlled or measured. Examples of such loads include wave loads on offshore structures, wind loads on buildings and traffic loads on bridges in such cases it is an advantage just to measure the natural (or ambient) responses and then estimate the modal parameters by performing an output-only modal identification. For civil structures the technique is often referred to as ambient response testing and ambient response analysis.
Application of output-only modal identification instead of traditional modal identification gives the user some clearly defined benefits in case of large structures and natural loading. Rather than loading the structure artificially and considering the natural loading as an unwanted noise source, the latter is used as the loading source. The main advantages of this technique are:
Testing is less time consuming since equipment for exciting the structure is not needed.
Testing does not interrupt the operation of the structure.
The measured response is representative of the real operating conditions of the structure.
When performing output-only modal identification of a structure, the user can perform the identification in the time domain or in the frequency domain. For output only identification, the time domain techniques have been rather dominating since no accurate techniques for frequency domain identification exists. However, since the frequency domain supports the physical intuition of the system, i.e. the user can observe the spectral densities and, thus, directly have an idea of the modes of the system by regarding the spectral peaks, simple and rather approximate techniques have been widely accepted for preliminary analysis. The most well-known frequency domain technique is the so-called classical approach, also denoted the basic frequency method, or the peak picking method, where the user simply chooses one of the frequency lines in the spectrum at the appearing peak as the natural (resonance) frequency and then estimates the corresponding mode shape as one of the columns of the spectral density matrix.
The classical approach is based on simple signal processing using the Discrete Fourier Transform, and is using the fact that well-separated modes can be estimated directly from the spectral density matrix at the peak, as shown in by Julius S. Bendat and Allan G. Piersol in xe2x80x9cEngineering Applications of Correlation and Spectral Analysisxe2x80x9d, John Wiley and Sons, 1993.
Other implementations of the technique make use of the coherence between channels as described by A. J. Felber in xe2x80x9cDevelopment of a Hybrid Bridge Evaluation Systemxe2x80x9d, Ph.D. thesis, Department of Civil Engineering, University of British Columbia, Vancouver, Canada, 1993. The term channel is commonly used for the output data of a sensor.
The main advantage of the classical approach compared to other approaches, such as two-stage time domain identification technique, or the one-stage time domain identification techniques, for example the Stochastic Subspace Identification algorithms is its user-friendliness. It is fast, simple to use, and gives the user a xe2x80x9cfeelingxe2x80x9d of the data he or she is dealing with.
The two-stage time domain has been described by H. Vold, J. Kundrat, G. T. Rocklin, and R. Russel in xe2x80x9cA Multi-Input Modal Estimation Algorithm For Mini-Computerxe2x80x9d, SAE Technical Paper Series, No. 820194, 1982; by S. R. Ibrahim and E. C. Milkulcik in xe2x80x9cThe Experimental Determination of Vibration Test Parameters From Time Responsesxe2x80x9d, The Shock and Vibration Bulletin, Vol. 46, No. 5, 1976, pp. 187-196; and by J.-N. Juang and R. S. Papa in xe2x80x9cAn Eigensystem Realization Algorithm For Modal Parameter Identification And Modal Reductionxe2x80x9d, J. of Guidance, Control and Dynamics, Vol. 8, No. 5, 1985, pp. 620-627. The Stochastic Subspace Identification algorithms is described by P. Van Overschee and B. De Moor in xe2x80x9cSubspace Identification for Linear Systemsxe2x80x9d, Kluwer Academic Publishers, 1996,
The classical technique gives reasonable estimates of natural frequencies and mode shapes if the modes are well-separated. However, in the case of close modes, it can be difficult to detect the close modes, and even in the case where close modes are detected, estimates becomes heavily biased by simple estimation of the mode shapes from one of the columns of the spectral matrix.
Further, the frequency estimates are limited by the frequency resolution of the spectral density estimate, and in all cases, damping estimation is uncertain or impossible.
It is the purpose of the invention to provide a frequency domain method for vibration analysis, i.e. output-only modal analysis, without these disadvantages, but where the user-friendliness is preserved.
This purpose is achieved by a method as mentioned by way of introduction and characterised as described in the characterising part of claim 1.
The invention significantly reduces the uncertainty in the estimation of vibrational modes of an object. Due to its user friendliness and fast obtainable results, the invention is a substantial improvement for output-only modal analysis, where the only major difference between modal parameters estimated from traditional modal testing and output-only modal analysis is that the output-only modal analysis yields unscaled mode shapes.
In the invention, it is assumed that the object has been excited over a broad frequency range by a signal, which has the same intensity at all frequencies. This kind of excitation is called white noise. As a consequence of the assumption with regard to the input excitation as white noise, the analysis according to the invention is directed to the response of the object and uses therefore the well-known term called output-only modal analysis.
A number of techniques are used in the invention. One is the so called Frequency Domain Decomposition (FDD) which is known from, among others, traditional modal analysis, where the structure is loaded by a known input However, in the case of traditional modal analysis the system matrix that is decomposed is not the spectral density matrix describing the responses, but the frequency response function (FRF) matrix relating input and output of the system.
As a second technique, the present invention uses the so-called Singular Value Decomposition (SVD) to perform the frequency domain decomposition of the spectral matrix. The SVD is known from numerical mathematics and is used in a number of different applications. However, the main application of this decomposition is to determine the rank of a matrix. In traditional modal analysis, the technique is mainly used to find the number of modes, but can also be used to estimate modal properties as described in Shih, C. Y., Y. G. Tsuei, R. J. Allemang and D. L. Brown: xe2x80x9cComplex Mode Indicator Function and its Applications to Spatial Domain Parameter Estimationxe2x80x9d, Mechanical Systems and Signal Proc., Vol. 2, No. 4, 1988 and in Shih, C. Y., Y. G. Tsuei, R. J. Allemang and D. L. Brown: xe2x80x9cA Frequency Domain Global Parameter Estimation Method for Multiple Reference Frequency Response Measurementsxe2x80x9d, Mechanical Systems and Signal Proc., Vol. 2, No. 4, 1988.
A third technique applied in the analysis of singular value decomposed spectral density functions is the modal assurance criterion (MAC) which is known from traditional modal analysis and described in Allemang, R. J. and D. L. Brown: xe2x80x9cA Correlation Coefficient for Modal Vector Analysisxe2x80x9d, Proc of the 1st International Modal Analysis Conference, IMAC, Orlando, Fla., USA, 1982. In the present invention it is used to isolate auto spectral density functions of the individual modes.
In order to estimate vibrational modes of an object, measurements are performed in a vibrational excited object using a number of spatially distributed vibration sensitive detectors. For example, a building can be exposed to a broad range of excitation frequencies due to wind or traffic, justifying the assumption of excitation by white noise.
At suitable predetermined locations, vibration sensitive detectors, for example accelerometers, are attached to the object and their response is measured and transformed to data which are stored in a suitable medium, for example a computer memory.
Those data are subject to further analysis, where the initial step is to achieve a spectrum equivalent to a spectral density function. This spectral density function is decomposed into auto spectral densities by a technique involving the modal assurance criterion (MAC), where the auto spectral densities can be interpreted as corresponding to independent vibrational modes. Then, the auto spectral densities are transformed from frequency domain to time domain in order to estimate damping and more accurate natural frequencies.
Though the method itself in practice is relatively simple to use leading to results in a fast way without the need of cumbersome calculations, the theory behind the method is not obvious and, therefore, one of the major achievements of the invention. Only a thorough theoretical treatment of the mathematical problem behind the theory leads to the method according to the invention, which is the reason for a detailed explanation of the theory in the following. After the theoretical approach, a practical approach of the method according to the invention will be given followed by examples for illustration.
When measuring the response of a structure, the response is sampled, it means that the signal is observed at discrete times, each of the observation times is spaced in time with the sampling interval xcex94t, and usually a number of observations of the response is obtained in a set of spatially distributed locations on the structure. When estimating the spectral density of the response, the spectral density is observed in a frequency band from zero frequency to the half of the sample frequency fs=1/xcex94t. In this frequency band the response of the structure is influenced by the long on the structure and by the dynamic modes in that frequency interval. This means, that the spectral matrix of the response of a structure is a combination of the responses of the modes that is present in the observed frequency band. In the following, it is shown that taking the Singular Value Decomposition (SVD) of the spectral matrix, the spectral matrix is decomposed into a set of auto spectral density functions, each corresponding to a single degree of freedom (SDOF) system. Each SDOF system corresponds to a certain mode of the system, i.e., instead of dealing with the more complicated combination of all modes and all the different spectral densities between the responses of the different locations of observation on the structure, the problem is reduced to dealing with auto spectral densities of the modal responses of the system. This result is exact in the case where the loading is white noise, the structure is lightly damped, and when the mode shapes of close modes are geometrically orthogonal. If these assumptions are not satisfied, the decomposition into SDOF systems is approximate, but still the results are significantly more accurate than the results of the classical frequency domain approach The SDOF auto spectral densities are identified using the known modal assurance criterion (MAC).
Obtaining a set of data from one of the detectors attached to the object of interest yields a set of time related measurements {(y1,t1), (y2,t2), . . . (yn,tn)}, where yn is the detector measurement at time tn. These measurements are in the following called the system response. The unknown excitation that has generated this system response is in the following denoted xn.
Determining the Spectral Density Function
From this set of measured system response, a frequency dependent spectral density function can be obtained by Fast Fourier Transform (FFT), which is a traditional approach. Assume that the system response has been measured at m locations on the object yn is then an mxc3x971 vector. Also assume that the Gaussian white noise excitation has been applied in r locations, which imply that xn is a rxc3x971 vector.
Due to the assumptions that the unknown excitation xn is Gaussian white noise, the mxc3x97m spectral density function Gyy(ixcfx89) of the system response will in the following be defined as
xe2x80x83Gyy(ixcfx89)=H(ixcfx89)GxxH(ixcfx89)Hxe2x88x92∞ less than xcfx89 less than ∞xe2x80x83xe2x80x83(1)
H(ixcfx89) is the mxc3x97r Frequency Response Function (FRF) matrix that relate the applied Gaussian white noise excitation to the system response. The superscript H denotes the Hermitian transpose, i.e. complex conjugate and transpose. The spectral density function Gxx of the Gaussian white noise excitation is a constant matrix indicating that all frequencies have been equally excited. The spectral density function reflects the predominant vibration frequencies xcfx89 that occur as peaks in the spectral density function. Since Gxx is a real and symmetric matrix Gyy(ixcfx89) will always be Hermitian, i.e. the elements below the diagonal is the complex conjugates of the corresponding elements above the diagonal.
However, this method when applied to measured data suffers from the introduction of the so-called xe2x80x9cleakagexe2x80x9d biasing in the density spectrum due to the assumption of periodic data when performing the FFT. This biasing results in a flattening of the before mentioned peaks in the spectral density function. To avoid this biasing, it is possible to calculate the covariance function for the data set first before a Fast Fourier Transform. To obtain an unbiased estimate of the covariance function however, a tedious calculation is involved.
Therefore, another method for spectral estimation is proposed and application of this method together with above mentioned frequency domain decomposition is a part of the invention. The method that is used is a combination of traditional FFT and a technique denoted Random Decrement Transform (RDT). First, a covariance function is estimated using the RDT technique, and then, the covariance function is transformed to frequency domain by the FFT. Since the covariance function is a decaying function in time, it is well known that by performing FFT on this covariance function instead of the data itself, the leakage bias of the resulting spectral density function will be significantly decreased. In fact, if the covariance function has decreased to zero at half of the maximum time lag, then the leakage bias is exactly zero.
The advantages of using the RDT technique are:
Instead of performing time consuming multiplication, as is necessary for traditional covariance function estimation, the RDT uses only additions which are much faster calculations for normal computers;
Instead of being constrained by being forced to use all the signal in one way only, the user can weight the different parts of the time signal by using different kinds of trigger level, where the trigger level is explained in the following below;
Instead of getting only one estimate of the covariance function, the user gets two independent estimates, one for the covariance function itself, and one for its derivative.
It has to be mentioned here that modes at higher frequencies are weighted more, if the covariance function in the above method is replaced by the derivative of the covariance function. Therefore, in some cases, using the derivative of the covariance function can be an advantage.
When the user of the RDT technique is to perform an estimate of the covariance function, he has to choose a so-called trigger criterion. The trigger criterion can for instance be that the signal is within a certain intensity range. Then, if the signal is within this range, the technique selects the data around the time where the trigger criterion is satisfied, and incorporates these data in the estimation process. Thus, the user has the possibility to estimate one covariance function for low amplitudes and one for large amplitudes. This is of importance in cases where the user might want to investigate whether the structure under consideration has been subject to vibrations with such large amplitudes, that the structural response has been become non-linear, i.e. parts of the structure has been yielding (steel) or cracking (concrete or masonry). In this case, if the structural response is non-linear, the user will observe a difference of the spectral density function estimates for the low and for the high amplitudes.
The RDT technique is a well known technique for estimation of system time functions. However, it is not common to transform the time functions to frequency domain, and the above mentioned advantages by doing so are not known.
Once, the spectral density function has been obtained the following tools are applied.
Theoretical Background of Frequency Domain Decomposition
Assume that the spectral density function Gyy(ixcfx89) of the system response has been estimated at discrete frequencies xcfx89=xcfx89j, for xcfx89jxe2x89xa70. Gyy(ixcfx89) is then described by a set of spectral density matrices, one for each of the discrete frequencies.
Consider the Hermitian spectral density matrix for xcfx89=xcfx89j. This matrix is then decomposed by taking the Singular Value Decomposition (SVD) of the matrix
Gyy(ixcfx89j)=UjSjUjH=sj1uj1uj1H+sj2uj2uj2H+, . . . , +sjmujmujmH0xe2x89xa6xcfx89xe2x89xa6∞xe2x80x83xe2x80x83(2)
where the matrix Uj=└uj1,uj2, . . . , ujm┘ is a unitary matrix holding the orthogonal singular vectors ujk as its columns, and Sj is a diagonal matrix holding the scalar singular values sjk. The singular values and corresponding vectors are always sorted so that the sj1 always will be the largest. This decomposition is performed for all the estimated spectral density matrices.
Near a peak, corresponding to one mode, the Eigenvalue of that mode becomes dominating resulting in a loss of rank of the spectral matrix at that particular frequency. This loss of rank is detected by the SVD by a similar loss of singular values having values significantly different from zero. If only one mode is completely dominating at a specific frequency xcfx89=xcfx89j, only sj1 will be significantly different from zero. As shown below, an estimate of the mode shape of this dominating mode is given by the corresponding singular vector uj1 assuming that the mode is lightly damped. The mode shape is the observable part of the Eigenvector that is associated with the dominating Eigenvalue.
If two close modes are dominating at xcfx89=xcfx89j then sj1 and sj2 will both be significant different from zero and corresponding mode shapes are given by the singular vectors uj1 and uj2 assuming that the mode shapes are orthogonal as uj1 and uj2 by definition are. This approach can in principle be extended to as many close modes as there are measurement channels m.
It is only at the peak of a dominating mode that uj1 will be the mode shape estimate. When moving away from the peak, the influence of the Eigenvalues of the other modes start to interfere. This means that the singular vector that is an estimate of the mode shape now can be any one of the singular vectors. However, by using the so-called Modal Assurance Criterion (MAC) it is possible to track which of the singular vectors that currently matches the uj1 at the peak. From this tracking it is then possible to determine which singular values that relates to the mode at the peak, and in this way construct the auto spectral density function of the particular mode. Due to the presence of noise it might not be possible to track the corresponding singular vectors over the hole frequency range. In this case the untracked part of the auto spectral density function is simply set to zero.
That statement that the first singular vector is an estimate for the mode shape can be verified by the following proof
Let xe2x80x9cxe2x88x92xe2x80x9d and superscript T denote complex conjugate and transpose, respectively. The FRF can then be written in partial fractions, i.e. pole/residue form                               H          ⁡                      (                          ⅈ              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                                                            ∑                                  k                  =                  1                                n                            ⁢                                                R                  k                                                                      ⅈ                    ⁢                                          xe2x80x83                                        ⁢                    ω                                    -                                      λ                    k                                                                        +                                                            R                  _                                k                                                              ⅈ                  ⁢                                      xe2x80x83                                    ⁢                  ω                                -                                                      λ                    _                                    k                                                      ⁢                          xe2x80x83                        -            ∞                    ≤          ω          ≤          ∞                                    (        3        )            
where n is the number of modes. xcexk is the Eigenvalue (pole) and Rk is the mxc3x97r residue matrix
xe2x80x83Rk=xcfx86kxcex3kTxe2x80x83xe2x80x83(4)
where xcfx86k,xcex3k is the mode shape vector and the modal participation vector, respectively. By inserting (3) into (1) the following relation is obtained                                           G            yy                    ⁡                      (                          ⅈ              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                                                            ∑                                  k                  =                  1                                n                            ⁢                                                ∑                                      s                    =                    1                                    n                                ⁢                                                      [                                                                                            R                          k                                                                                                      ⅈ                            ⁢                                                          xe2x80x83                                                        ⁢                            ω                                                    -                                                      λ                            k                                                                                              +                                                                                                    R                            _                                                    k                                                                                                      ⅈ                            ⁢                                                          xe2x80x83                                                        ⁢                            ω                                                    -                                                                                    λ                              _                                                        k                                                                                                                ]                                    ⁢                                                                                    G                        xx                                            ⁡                                              [                                                                                                            R                              s                                                                                                                      ⅈ                                ⁢                                                                  xe2x80x83                                                                ⁢                                ω                                                            -                                                              λ                                s                                                                                                              +                                                                                                                    R                                _                                                            s                                                                                                                      ⅈ                                ⁢                                                                  xe2x80x83                                                                ⁢                                ω                                                            -                                                                                                λ                                  _                                                                s                                                                                                                                    ]                                                              H                                                                        ⁢                          xe2x80x83                        -            ∞                    ≤          ω          ≤          ∞                                    (        5        )            
By multiplying the two partial fraction factors, and making use of the Heaviside partial fraction theorem, the spectral density function Gyy(ixcfx89) can, after some mathematical manipulations, be reduced to a pole/residue form as follows                                           G            yy                    ⁡                      (                          ⅈ              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                                                            ∑                                  k                  =                  1                                n                            ⁢                                                A                  k                                                                      ⅈ                    ⁢                                          xe2x80x83                                        ⁢                    ω                                    -                                      λ                    k                                                                        +                                                            A                  _                                k                                                              ⅈ                  ⁢                                      xe2x80x83                                    ⁢                  ω                                -                                                      λ                    _                                    k                                                      +                                          B                k                                                              -                  ⅈω                                -                                  λ                  k                                                      +                                                            B                  _                                k                                                                                  -                    ⅈ                                    ⁢                                      xe2x80x83                                    ⁢                  ω                                -                                                      λ                    _                                    k                                                      ⁢                          xe2x80x83                        -            ∞                    ≤          ω          ≤          ∞                                    (        6        )            
where Ak and Bk are the k""th residue matrices of the spectral density function Gyy(ixcfx89). The contribution from the Bk elements is much smaller than the contribution from the Ak elements and will in the following be regarded as negligible.
As the spectral density function Gyy(ixcfx89) itself the residue matrices are mxc3x97m Hermitian matrices.
The residue matrix Ak is given by                               A          k                =                              R            k                    ⁢                                                    G                xx                            ⁡                              (                                                                            ∑                                              s                        =                        1                                            n                                        ⁢                                                                  R                        s                                                                                              -                                                                                    λ                              _                                                        k                                                                          -                                                  λ                          s                                                                                                      +                                                                                    R                        _                                            s                                                                                      -                                                                              λ                            _                                                    k                                                                    -                                                                        λ                          _                                                s                                                                                            )                                      H                                              (        7        )            
The contribution to the residue from the k""th mode is given by                               A          k                =                                            R              k                        ⁢                          G              xx                        ⁢                          R              k              H                                            2            ⁢                          (                              2                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  f                  k                                ⁢                                  ζ                  k                                            )                                                          (        8        )            
where the denominator of (8) is two times minus the teal part of the Eigenvalue xcexk=xe2x88x922xcfx80fkxcex6k+i2xcfx80fk{square root over (1xe2x88x92xcex6k2)} with fk,xcex6k being the natural frequency and damping ratio, respectively. As it appears, this term becomes dominating when the damping is light, i.e. when xcex6k tends to zero. Thus, is case of fight damping, the residue becomes proportional to the mode shape vector
Akxe2x88x9dRkGxxRkH=xcfx86kxcex3kTGxx{overscore (xcex3)}kxcfx86kH=dkxcfx86kxcfx86kHxe2x80x83xe2x80x83(9)
where dk is a scalar constant that is non-negative since Gxx always will be at least semi-definite.
At a certain frequency xcfx89, only a limited number of modes will contribute significantly, typically one or two modes. Let this set of modes be denoted by Sub(xcfx89). Thus, in the case of a lightly damped object, the spectral density function Gyy(ixcfx89) can always be written                                           G            yy                    ⁡                      (                          ⅈ              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                                                            ∑                                  k                  ∈                                      Sub                    ⁡                                          (                      ω                      )                                                                                  ⁢                                                                    d                    k                                    ⁢                                      ϕ                    k                                    ⁢                                      ϕ                    k                    H                                                                                        ⅈ                    ⁢                                          xe2x80x83                                        ⁢                    ω                                    -                                      λ                    k                                                                        +                                                                                d                    _                                    k                                ⁢                                                      ϕ                    _                                    k                                ⁢                                  ϕ                  k                  T                                                                              ⅈ                  ⁢                                      xe2x80x83                                    ⁢                  ω                                -                                                      λ                    _                                    k                                                      ⁢                          xe2x80x83                        -            ∞                    ≤          ω          ≤          ∞                                    (        10        )            
Now (10) describes the spectral density function from xe2x88x92∞ to ∞. However, in practice, only the positive part of the spectral density function is considered, which reduces (10) to                                           G            yy                    ⁡                      (                          ⅈ              ⁢                              xe2x80x83                            ⁢              ω                        )                          =                                            ∑                              k                ∈                                  Sub                  ⁡                                      (                    ω                    )                                                                        ⁢                                                            d                  k                                ⁢                                  ϕ                  k                                ⁢                                  ϕ                  k                  H                                                                              ⅈ                  ⁢                                      xe2x80x83                                    ⁢                  ω                                -                                  λ                  k                                                              =                                                    ∑                                  k                  ∈                                      Sub                    ⁡                                          (                      ω                      )                                                                                  ⁢                                                                    d                    k                                                                              ⅈ                      ⁢                                              xe2x80x83                                            ⁢                      ω                                        -                                          λ                      k                                                                      ⁢                                  ϕ                  k                                ⁢                                  ϕ                  k                  H                                ⁢                                  xe2x80x83                                ⁢                0                                      ≤            ω            ≤            ∞                                              (        11        )            
This is a modal decomposition of the spectral matrix. The expression is similar to the results one would get directly from (2) under the assumption of independent white noise input, i.e. a diagonal spectral input matrix. For each frequency and each eigenvalue, value, the ratio in front of the mode shape product in (11) will be a positive constant that can always be made real by proper scaling of the corresponding mode shapes.
Identification Algorithm
Near a peak corresponding to the k th mode in the spectrum say at xcfx89=xcfx89j, this mode or maybe a possible close mode will be dominating. If only the k th mode is dominating, there will only be one term in (11). Thus, in this case, the first singular vector uj1 is an estimate of the mode shape
{circumflex over (xcfx86)}=uj1xe2x80x83xe2x80x83(12)
and the corresponding singular value is the current value of auto power spectral density function of the corresponding single degree of freedom system at that specific frequency, refer to (11).
Define the MAC value between two vectors as                               MAC          ⁡                      (                                          ϕ                ^                            ,                              u                nm                                      )                          =                              "LeftBracketingBar"                                                            ϕ                  ^                                H                            ⁢                              u                nm                                      "RightBracketingBar"                                                                                                    ϕ                    ^                                    H                                ⁢                                  ϕ                  ^                                                      ⁢                                                            u                  nm                  H                                ⁢                                  u                  nm                                                                                        (        13        )            
This value describes the correlation between the two vectors. If the vectors are similar except for a constant scaling the MAC value is 1. If they are complete orthogonal the value will be 0.
Using the MAC value in (13) the rest of the auto spectral density function is identified around the peak by comparing the mode shape estimate {circumflex over (xcfx86)} in (12) with the singular vectors for the frequency lines around the peak. As long as a singular vector unm is found that has MAC value near 1 with {circumflex over (xcfx86)}, the corresponding singular value belongs to the auto spectral density function of that specific mode
If at a certain distance firm the peak of the mode none of the singular values has a singular vector with a MAC value larger than a certain limit value xcexa9, the search for matching parts of the auto spectral density function is terminated. The remaining unidentified part of the auto spectral density function is set to zero. From this fully or partially identified auto spectral density function of the SDOF system, the natural frequency and the damping are obtained by taking the auto spectral density function back to time domain by inverse discrete Fourier transform.
From the free decay time domain function, which is also the auto correlation function of the SDOF system of the k th mode, the natural frequency and the damping ratio is found by estimating crossing times and logarithmic decrement First, all extremes rn, both peaks and valleys, on the auto correlation function are found. The logarithmic decrement xcex4 is then given by                     δ        =                              2            n                    ⁢          ln          ⁢                      xe2x80x83                    ⁢                      (                                          r                0                                            "LeftBracketingBar"                                  r                  n                                "RightBracketingBar"                                      )                                              (        14        )            
where r0 is the initial value of the auto correlation function and rn is the n th extreme. Thus, the logarithmic decrement and the initial value of the correlation function can be found by linear regression on nxcex4 and 2ln(|rn|), and the damping ratio is given by the well known formula                               ζ          k                =                  δ                                                    δ                2                            +                              4                ⁢                                  xe2x80x83                                ⁢                                  π                  2                                                                                        (        15        )            
A similar procedure is adopted for determination of the natural frequency. The natural frequency is found by making a linear regression on the crossing times and the times corresponding to the extremes and using that the damped natural frequency fkd and the undamped natural frequency fk is related by                               f          k                =                              f            k            d                                              1              -                              ζ                k                2                                                                        (        16        )            
The extreme values and the corresponding times are found by quadratic interpolation, whereas the crossing times where found by linear interpolation.